John Bercow, the Commons Speaker, suffered a significant setback on Thursday night when Jack Straw and Dame Margaret Beckett became the most senior figures at Westminster to raise questions about plans to appoint an Australian parliamentary official to the most senior unelected post at Westminster.

The two former Labour foreign secretaries, who served as leaders of the House of Commons under Tony Blair, are supporting a cross-party campaign calling for Carol Mills to face a confirmation hearing before a parliamentary select committee.

Bercow is facing intense pressure over the decision of a six-strong cross-party panel to appoint Mills, secretary of the department of parliamentary services in Canberra, as clerk and chief executive of the Commons. MPs on all sides of the house believe that, while Mills has strong management experience, she has less experience of overseeing parliamentary procedure and guarding MPs' privileges.

The concerns at Westminster were fuelled by a warning from Rosemary Laing, the clerk of the Australian senate, who said Mills has a record of running "ancillary services" but little experience of overseeing parliamentary procedures.

Bercow is understood to be determined to press ahead with the appointment of Mills on grounds that she was chosen by a panel of five senior MPs, plus the parliamentary ombudsman, Dame Julie Mellor, who held two lengthy interviews with the candidates. The Speaker's office has passed the name of Mills to Downing Street for onward passage to the palace for approval by the Queen. Downing Street is said to be concerned by the strength of opposition to the proposed appointment. But No 10 officials are indicating that David Cameron believes it would be inappropriate for him to intervene.

Opponents believe their best hope of blocking Mills is for No 10 to delay sending her name to the palace while the Australian senate's privileges committee carries out an inquiry into allegations that officials in her office used footage from CCTV cameras to monitor the office of a senator. The cameras were already in place, in line with the rules, and the officials were not monitoring the senator but a parliamentary official.

The Speaker's opponents also believe that subjecting Mills to a pre-appointment hearing will show whether she has the correct qualifications for what is described as one of the most senior jobs in Britain's unwritten constitutional settlement, as adviser on the procedures of parliament to the speaker, the government and the opposition.

Straw told the Guardian he supported the confirmation hearings, though he is making no judgment about Mills. "Given the controversy, and without making any observations about the relative merits of the candidate, I think that such pre-appointment scrutiny would be a good way of resolving this."

Beckett echoed Straw's call for a confirmation hearing as she raised concerns that Mills was being lined up to focus mainly on the role of chief executive of the Commons rather than clerk. She said: "These days you cannot make an appointment like this without select committee scrutiny. The idea of having a chief executive role, which is not the clerk to the house, has often been discussed in the past.

"But as far as I know the house has never decided actually to go down that road. It is a step of perhaps major constitutional significance as far as the house as a whole is concerned."

In an email to Sir Robert Rogers, who retires as clerk of the Commons at the end of this month, the clerk to the Australian senate, Rosemary Laing, wrote: "We were utterly taken aback here when we saw a brief press report in early July that Carol Mills had emerged as 'frontrunner' to take over from you, and have followed events with increasing disbelief and dismay. It seemed to us impossible that someone without parliamentary knowledge and experience could be under consideration for such a role."

The interventions by Straw and Beckett came after the former Tory policing minister Damian Green said he was "extremely disturbed" by the parallels between the alleged breach of parliamentary privilege in Canberra and a police raid on his office at Westminster in 2008. Green called for a delay in the appointment of Mills until an Australian senate committee has conducted an inquiry into the use of CCTV cameras by officials in her department to monitor the office of the veteran Labor senator John Faulkner as part of an investigation into a parliamentary official.

Green, whose parliamentary office was raided by police in 2008 after a senior official at Westminster failed to alert the clerk of the Commons, said he would be contacting the Speaker to raise concerns about the parallels between the two cases.